Yesterday afternoon there was an event at CUNY featuring a panel discussion on Chern-Simons terms. Nothing new there, although it was interesting to hear first-hand from Witten the story of how he came up with the Chern-Simons-Witten theory. One piece of news I heard from Nikita Nekrasov was that he was missing a talk that day at the Simons Center in Stony Brook by Maxim Kontsevich, who would be arguing that the Hodge and Tate conjectures were not true. The video of that talk has now appeared, see here.
I’m way behind in preparing for my class for tomorrow, so haven’t had time to watch the full video and ask experts about it. Will try and learn more tomorrow after my class, but it does seem that if Kontsevich is right that would be a dramatic development. If you are able to evaluate Kontsevich’s arguments, any comments welcome. Tomorrow I’ll also try and at least find some good references to suggest for anyone who wants to learn the background of what these conjectures say.
Update: I see there’s an older version of this idea described here.
Update: Having watched the video and talked to a few people about it, I fear that there’s not much new here, and nothing likely to convince experts that a falsification of the Hodge or Tate conjectures is on the horizon. Kontsevich himself introduces the talk as “not really a talk, but a kind of after-dinner rant.” He for a long time has been trying to find examples that could falsify the Hodge conjecture, with no success so far, and from what I can tell, he doesn’t have a new compelling proposal for where to look and how to do this.
In his earlier talk, Kontsevich suggested looking for a counterexample to the Hodge conjecture on abelian fourfolds of Weil type. However, Markman has proved the Hodge conjecture for abelian fourfolds of Weil type with determinant 1, so this may not be such a great idea.
Markman, Eyal The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians. J. Eur. Math. Soc. (JEMS) 25 (2023), no. 1, 231–321.
For those of us among the algebraici geometriae ignorami, I recommend:
“The Tate conjecture over finite fields (AIM talk)” by J. S. Milne, 2007, somewhat revised and updated in 2021
“Known cases of the Hodge conjecture” by Genival da Silva Jr, 2021
V. Kumar Murty’s 1986 book
Was the panel filmed?
I believe so, and I think the whole thing was on Zoom. I don’t know if there are any plans to make the video public. If so, one place it might appear is at the Einstein Chair Mathematics Seminar Youtube site here
A little history. What’s interesting about abelian varieties of Weil type is that they have cohomology classes, predicted to be algebraic by the Hodge conjecture, but not obviously algebraic (not in the Q-algebra generated by the divisor classes). When Weil constructed the varieties, he wrote to Tate: “As you and Mumford seem to believe Hodge’s conjecture, it is now up to you to exhibit algebraic cycles corresponding to these abnormal classes. As I incline to disbelieve it, I shall rather attempt to show that there is no such cycle; but I have no idea how this could be done.”
I think Kontsevich’s points are important and good to think about carefully even if one doesn’t share his strong opinion about the Hodge conjecture. The philosophy of motives has a few components which can be considered independently:
0) There are many cohomology theories for algebraic varieties (singular over C, de Rham, l-adic, crystalline…) with very different constructions and nevertheless striking similarities. Moreover, these similarities underlie interesting phenomena in algebraic geometry, arithmetic geometry, number theory, mathematical physics…
1) There should exist a universal cohomology theory, valued in some abelian category of motives, which explains these similarities. This abelian category should be a rigid Tannakian category (with the tensor product related to products of varieties and the Künneth formula, and rigidity coming from Poincaré duality), and the corresponding “motivic Galois groups” should reduce some questions about these similarities to tractable questions in the representation theory of pro-algebraic groups. There should be a subcategory of pure motives (for motives of smooth projective varieties) embedded in the larger category of (all, mixed) motives, together with a functorial weight filtration with pure graded objects, so that every motive is canonically an iterated extension of pure motives. This subcategory of pure motives should be the category of representations of the maximal pro-reductive quotient of the motivic Galois group.
2) The category of motives should be constructed solely from algebraic geometry, and morphisms groups in that category should be ultimately describable in terms of algebraic cycles: directly via Grothendieck’s original proposal for pure motives, and more indirectly (in terms of “higher Chow groups”/higher algebraic K-theory) for mixed motives.
One way to summarize Kontsevich’s talk is that he believes in 0) and in some form of 1) but not in 2).
Point 0) is essentially an “empirical” observation which has been apparent and pretty uncontroversial ever since Grothendieck, and has only become clearer since then.
Point 1) still seems like the most parsimonious explanation for 0). In his talk, Kontsevich indicates that he does believes in some version of 1). In characteristic 0, mathematicians have actually constructed abelian/Tannakian categories which realise some of these desiderata (Deligne’s pure motives based on absolute Hodge cycles, André’s pure motives based on motivated cycles, Nori’s mixed motives with an important alternative construction by Ayoub and Choudhury-Gallauer). However, we we understand very little about morphism groups in these categories, because they are built from the choice of an embedding of your base field into C and transcendental constructions involving singular cohomology, and not a priori related to algebraic geometry. We also essentially don’t know what to do in positive characteristic. Consequently, these constructions only have had a few applications so far.
Point 2) is what that Kontsevich rejects. As far as I can tell, the standard heuristic argument for believing in 2) is that algebraic cycles are a clean mechanism for forming bridges between different cohomology theories: namely, two cohomology classes in different cohomology theories can be the cycle class of the same algebraic cycle! So if algebraic cycles with prescribed cohomological properties were as abundant as the Hodge and Tate conjecture (as well as a few other key conjectures about cycles like the Beilinson-Soulé and Bloch-Beilinson-Murre conjectures) predict, we could indeed construct the categories predicted by 1) purely geometrically. This was explained by Grothendieck in the pure case (where one only needs the standard conjectures), and would follow from Voevodsky’s theory in the mixed case, as I discuss a little below.
Sidenote: the full force of the Hodge/Tate conjecture is not needed, one “only” needs Grothendieck’s standard conjectures; personally, I would not be too sad if the Hodge conjecture was false as long as the standard conjectures are ok! But Kontsevich also rejects that possibility; as shown by André, the standard conjectures for all smooth projective complex varieties imply the Hodge conjecture for abelian varieties, and Kontsevich is looking for counterexamples to the Hodge conjecture on abelian varieties.
As Kontsevich stresses, the evidence for all these cycle conjectures is quite thin on the ground. There are simple situations like Weil-type abelian varieties where we don’t know where cycles lifting certain natural Hodge classes could come from (even with recent great progress by Markman), and almost all known cases rely on cleverly bootstrapping existing cycles and not on finding new general principles of construction of cycles. I think Kontsevich is right that some of us “motivists” have become too trusting in the general web of motivic conjectures, and that more effort should go into stress-testing them by looking for counterexamples as well as new invariants which could lead to counterexamples (like his tropical proposal).
However, if 2) is false and 1) is true, then to make 1) usable we would still need an explanation of where the morphism groups of the abelian categories in 1) come from, if not from algebraic geometry! Especially in positive characteristic where we do not have the crutch of singular cohomology and we do not understand why something like independence of l-adic Betti numbers for general varieties, which would follow from any reasonable version of 1), should be true.
An interesting part of Kontsevich’s discussion is about the relative setting. As he notes, in the context say of topological Q-local systems over a smooth algebraic curve over a number field with some given complex embedding, we have some tentative criteria for what “local systems of pure motives” should be: integrality of monodromy, G-function /growth of power series coefficients of solution of the associated differential equation , vanishing p-curvatures of the differential equations in characteristic p, existence of Dwork-Frobenius structures. However, (outside of recent progress in some directions for rigid local systems on higher-dimensional varieties by Esnault-Groechenig) we don’t really know how to prove the equivalence of these very disparate conditions, other than to try to prove that they are all equivalent to “coming from smooth projective families of algebraic varieties”! A useful theory of 1) should include abelian categories of “motivic local systems” and needs to explain why these conditions all seem to be equivalent; again, if not by geometry, how?
To conclude, let me clarify the position of the work of Voevodsky in this picture. Voevodsky has, among many other things, constructed triangulated categories which look like the *derived categories* of the categories predicted in 1), have a purely algebro-geometric construction, and whose morphism groups can be described (by some very non-trivial theorems) in terms of algebraic cycles. This has had some nice applications and also revealed a wider picture of “motivic homotopy theory” which is interesting independently of the validity of points 1)-2). However, finding an abelian category of motives inside of these triangulated categories (“constructing the motivic t-structure”) is known, at least in characteristic 0, to be essentially equivalent to the same hard conjectures about algebraic cycles.
I should also point out Ayoub has proposed a very interesting approach to the motivic t-structure on Voevodsky motives which tries to bypass the conjectures on cycles, and which so far has lead to the new construction of Nori motives mentioned above and to a functional/geometric version of the Grothendieck-Kontsevich-Zagier conjecture on periods of varieties over number fields- ironically, one of the many conjectures that Kontsevich now believes to be false…
jsm and Simon Pepin Lehalleur,
Many thanks for the informative comments.
Bear in mind that it is commonly said that Deligne was skeptical of the Hodge conjecture and tried to find counterexamples on abelian varieties, but failing that, developed the theory of absolute Hodge cycles. This tropical speculation has been around for at least ten years. I once tried to get Katzarkov to explain to me why tropical geometry should succeed where Deligne failed, but couldn’t get a reasonable answer. Someone should push Kontsevich hard on this until he says something that other people can understand.